A comparison between two de Rham complexes in diffeology

Abstract

There are two de Rham complexes in diffeology. The original one is due to Souriau and another one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech-de Rham spectral sequence in diffeology. In particular, a comparison map enables us to conclude that the first singular de Rham cohomology for the irrational torus T θ is isomorphic to the direct sum of the original one and the group of equivalence classes of flow bundles over T θ with connection 1-forms.